The last and only remaining item to get the 3562A with the defective A3 ROM Board (03562-66503) back into service, replacing the defective EPROM. Well, I thought this would be a 30 minute job, but it ended up taking a few hours longer. Why? Multiple reasons:

(1) A bug in the AVR eprom reader software, specifically written to read the 3562A rom board (and similar boards, or other instruments – software always needs some adaption). One of the address lines was not toggled-ended up with corrupted data read from the “good” rom board.
Learning for today: always check the EPROM data read for validity, by checking for repeat patters, and by reviewing all the byte values. There should be at least a few 0xff values, otherwise, one of the data bus lines may be broken.

(2) Turns out, there are two versions even of the Rev B rom board, same part number, but eproms U118 and U218 that have the checksums are different. So, needed to desolder these two as well, and replace with the updated version from the working board.

The good ones on the left, the bad guys on the right…

(3) With all this, my eprom programmer, the only one I have that can handle 27256 EPROM had a defective jumper! No contact on one of the address lines….

After all these efforts: that’s the board, after repair:

The replaced EPROMs are now in sockets – just in case, should they ever fail again.

Just in case you have to do a similar job – here are the EPROM images. Keep in mind, Rev B, 36x 27256 EPROM (sure, you can also use 27C256).
hp 3562a A3 ROM BOARD rev B

And, finally, let’s switch the power on-

All tests passed!

Note – just discovered, someone is offering a spare 3562A ROM board for about USD 100 on xbay…. well, well, but in the end, better a thorough repair, with all EPROM images captured, than just switching some board!